Find the largest interval for which the following initial value problems are guaranteed to have a unique solution. Explain your reasoning (5 points each). DO NOT SOLVE THE EQUATIONS.

Answer and Step-by-step explanation:

Solution:

The given equation is:

t (t -4)y” + 3ty’ + 4y = 2

Where,

Y (3) = 0, y’ (3) = -1

To find out the largest interval, which have unique solution,

Divide the given equation with t(t – 4):

Y” + 3y’ / t(t-4) + 4y / t(t-4) = 2 / t (t -4)

Solution exists on (0, 4)

Because:

P(t) = 3 / t -4, continuous on ( -∞, 4) and (4, ∞)

Q(t) = 4/t(t-4) , continuous on (-∞, 0), (0,4) and (4, ∞)

G(t) = 2 / t(t-4) continuous on (-∞, 0), (0,4) and (4,∞)

T0 = 3,

So solution exist on (0, 4)

Hence, (0, 4) is the largest interval for which the initial value problem:

t (t -4)y” + 3ty’ + 4y = 2

have a unique solution.